On what planet is that monster a sane gate driver transformer for a couple of TO247 IGBTs?
Also 400+KHz, with IGBTs? really, are you sure?
From the look of that scope plot you are probably measuring between something like local ground and the high side gate?
Yea, not going to get sane measurements that way (Quite apart from maybe blowing up the scope if you get it wrong). An isolated differential probe would be your friend, as with the whole dogs dinner up at rectified mains I don't see a reasonable way to get a trace of the actual gate drive voltage (as in the gate-emitter voltage, which is what matters).
You might also wish to consider the potential between the low side gates and the high side ones, and the possibility that two strands of cat 5 signal wire just might not be insulated for all that voltage, just a thought.
Can you get your schools technicians (Generally more useful the the lecturers for this kind of thing) to help you sort it?
This is impossible to debug remotely (Especially as you clearly don't really know how to measure what you need to measure) and you really need someone who has high voltage, high frequency power experience on site to try to unfuck that thing. Expect there to be new IGBTs (and possibly new IGBT over voltage protection silicon) in your future together with a sane gate transformer and probably a current transformer to protect the IGBTs, also expect there to be some sheet metal and an isolation transformer involved, together with some much better wiring.
You have clearly bitten off WAY more then you can chew here.
Regards, Dan.